MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fullres2c Unicode version

Theorem fullres2c 13813
Description: Condition for a full functor to also be a full functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
Hypotheses
Ref Expression
ffthres2c.a  |-  A  =  ( Base `  C
)
ffthres2c.e  |-  E  =  ( Ds  S )
ffthres2c.d  |-  ( ph  ->  D  e.  Cat )
ffthres2c.r  |-  ( ph  ->  S  e.  V )
ffthres2c.1  |-  ( ph  ->  F : A --> S )
Assertion
Ref Expression
fullres2c  |-  ( ph  ->  ( F ( C Full 
D ) G  <->  F ( C Full  E ) G ) )

Proof of Theorem fullres2c
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffthres2c.a . . . 4  |-  A  =  ( Base `  C
)
2 ffthres2c.e . . . 4  |-  E  =  ( Ds  S )
3 ffthres2c.d . . . 4  |-  ( ph  ->  D  e.  Cat )
4 ffthres2c.r . . . 4  |-  ( ph  ->  S  e.  V )
5 ffthres2c.1 . . . 4  |-  ( ph  ->  F : A --> S )
61, 2, 3, 4, 5funcres2c 13775 . . 3  |-  ( ph  ->  ( F ( C 
Func  D ) G  <->  F ( C  Func  E ) G ) )
7 eqid 2283 . . . . . . . 8  |-  (  Hom  `  D )  =  (  Hom  `  D )
82, 7resshom 13323 . . . . . . 7  |-  ( S  e.  V  ->  (  Hom  `  D )  =  (  Hom  `  E
) )
94, 8syl 15 . . . . . 6  |-  ( ph  ->  (  Hom  `  D
)  =  (  Hom  `  E ) )
109oveqd 5875 . . . . 5  |-  ( ph  ->  ( ( F `  x ) (  Hom  `  D ) ( F `
 y ) )  =  ( ( F `
 x ) (  Hom  `  E )
( F `  y
) ) )
1110eqeq2d 2294 . . . 4  |-  ( ph  ->  ( ran  ( x G y )  =  ( ( F `  x ) (  Hom  `  D ) ( F `
 y ) )  <->  ran  ( x G y )  =  ( ( F `  x ) (  Hom  `  E
) ( F `  y ) ) ) )
12112ralbidv 2585 . . 3  |-  ( ph  ->  ( A. x  e.  A  A. y  e.  A  ran  ( x G y )  =  ( ( F `  x ) (  Hom  `  D ) ( F `
 y ) )  <->  A. x  e.  A  A. y  e.  A  ran  ( x G y )  =  ( ( F `  x ) (  Hom  `  E
) ( F `  y ) ) ) )
136, 12anbi12d 691 . 2  |-  ( ph  ->  ( ( F ( C  Func  D ) G  /\  A. x  e.  A  A. y  e.  A  ran  ( x G y )  =  ( ( F `  x ) (  Hom  `  D ) ( F `
 y ) ) )  <->  ( F ( C  Func  E ) G  /\  A. x  e.  A  A. y  e.  A  ran  ( x G y )  =  ( ( F `  x ) (  Hom  `  E ) ( F `
 y ) ) ) ) )
141, 7isfull 13784 . 2  |-  ( F ( C Full  D ) G  <->  ( F ( C  Func  D ) G  /\  A. x  e.  A  A. y  e.  A  ran  ( x G y )  =  ( ( F `  x ) (  Hom  `  D ) ( F `
 y ) ) ) )
15 eqid 2283 . . 3  |-  (  Hom  `  E )  =  (  Hom  `  E )
161, 15isfull 13784 . 2  |-  ( F ( C Full  E ) G  <->  ( F ( C  Func  E ) G  /\  A. x  e.  A  A. y  e.  A  ran  ( x G y )  =  ( ( F `  x ) (  Hom  `  E ) ( F `
 y ) ) ) )
1713, 14, 163bitr4g 279 1  |-  ( ph  ->  ( F ( C Full 
D ) G  <->  F ( C Full  E ) G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149    Hom chom 13219   Catccat 13566    Func cfunc 13728   Full cful 13776
This theorem is referenced by:  ffthres2c  13814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-hom 13232  df-cco 13233  df-cat 13570  df-cid 13571  df-homf 13572  df-comf 13573  df-ssc 13687  df-resc 13688  df-subc 13689  df-func 13732  df-full 13778
  Copyright terms: Public domain W3C validator